Quantum Field Theory II:
Quantum Electrodynamics


Eberhard Zeidler

Quantum Field Theory II:
Quantum Electrodynamics
A Bridge between Mathematicians
and Physicists


Eberhard Zeidler
Max Planck Institute
for Mathematics in the Sciences
Inselstr. 22-26
04103 Leipzig
Germany

ISBN 978-3-540-85376-3

e-ISBN 978-3-540-85377-0

DOI 10.1007/978-3-540-85377-0
Library of Congress Control Number: 2006929535
Mathematics Subject Classification (2000): 35-XX, 47-XX, 49-XX, 51-XX, 55-XX, 81-XX, 82-XX
c 2009 Springer-Verlag Berlin Heidelberg
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TO FRIEDRICH HIRZEBRUCH
IN GRATITUDE


Preface

And God said, Let there be light; and there was light.
Genesis 1,3
Light is not only the basis of our biological existence, but also an essential
source of our knowledge about the physical laws of nature, ranging from
the seventeenth century geometrical optics up to the twentieth century
theory of general relativity and quantum electrodynamics.
Folklore
Don’t give us numbers: give us insight!
A contemporary natural scientist to a mathematician

The present book is the second volume of a comprehensive introduction to
the mathematical and physical aspects of modern quantum field theory which

comprehends the following six volumes:
Volume
Volume
Volume
Volume
Volume
Volume

I: Basics in Mathematics and Physics
II: Quantum Electrodynamics
III: Gauge Theory
IV: Quantum Mathematics
V: The Physics of the Standard Model
VI: Quantum Gravitation and String Theory.

It is our goal to build a bridge between mathematicians and physicists based
on the challenging question about the fundamental forces in
• macrocosmos (the universe) and
• microcosmos (the world of elementary particles).
The six volumes address a broad audience of readers, including both undergraduate and graduate students, as well as experienced scientists who want
to become familiar with quantum field theory, which is a fascinating topic in
modern mathematics and physics.
For students of mathematics, it is shown that detailed knowledge of the
physical background helps to motivate the mathematical subjects and to


VIII

Preface


discover interesting interrelationships between quite different mathematical
topics. For students of physics, fairly advanced mathematics are presented,
which is beyond the usual curriculum in physics. The strategies and the
structure of the six volumes are thoroughly discussed in the Prologue to
Volume I. In particular, we will try to help the reader to understand the
basic ideas behind the technicalities. In this connection, the famous ancient
story of Ariadne’s thread is discussed in the Preface to Volume I. In terms
of this story, we want to put the beginning of Ariadne’s thread in quantum
field theory into the hands of the reader.
The present volume is devoted to the physics and mathematics of
light.
It contains the following material:
Part I: Introduction
• Chapter 1: Mathematical Principles of Modern Natural Philosophy
• Chapter 2: The Basic Strategy of Extracting Finite Information from
Infinities – Ariadne’s Thread in Renormalization Theory
• Chapter 3: The Power of Combinatorics
• Chapter 4: The Strategy of Equivalence Classes in Mathematics
Part II: Basic Ideas in Classical Mechanics
• Chapter 5: Geometrical Optics
• Chapter 6: The Principle of Critical Action and the Harmonic Oscillator as a Paradigm
Part III: Basic Ideas in Quantum Mechanics
• Chapter 7: Quantization of the Harmonic Oscillator – Ariadne’s Thread
in Quantization
• Chapter 8: Quantum Particles on the Real Line – Ariadne’s Thread in
Scattering Theory
• Chapter 9: A Glance at General Scattering Theory.
Part IV: Quantum Electrodynamics (QED)
• Chapter 10: Creation and Annihilation Operators
• Chapter 11: The Basic Equations in Quantum Electrodynamics

• Chapter 12: The Free Quantum Fields of Electrons, Positrons, and
Photons
• Chapter 13: The Interacting Quantum Field, and the Magic Dyson
Series for the S-Matrix
• Chapter 14: The Beauty of Feynman Diagrams in QED
• Chapter 15: Applications to Physical Effects
Part V: Renormalization
• Chapter 16: The Continuum Limit
• Chapter 17: Radiative Corrections of Lowest Order
• Chapter 18: A Glance at Renormalization to all Orders of Perturbation
Theory
• Chapter 19: Perspectives


Preface

IX

We try to find the right balance between the mathematical theory and its
applications to interesting physical effects observed in experiments. In particular, we do not consider purely mathematical models in this volume.
It is our philosophy that the reader should learn quantum field theory
by studying a realistic model, as given by quantum electrodynamics.
Let us discuss the main structure of the present volume. In Chapters 1
through 4, we consider topics from classical mathematics which are closely
related to modern quantum field theory. This should help the reader to understand the basic ideas behind quantum field theory to be considered in
this volume and the volumes to follow. In Chapter 1 on the mathematical
principles of modern natural philosophy, we discuss
• the infinitesimal strategy due to Newton and Leibniz,
• the optimality principle for processes in nature (the principle of critical action)
and the calculus of variations due to Euler and Lagrange, which leads to the

fundamental differential equations in classical field theory,
• the propagation of physical effects and the method of the Green’s function,
• harmonic analysis and the Fourier method for computing the Green’s functions,
• Laurent Schwartz’s theory of generalized functions (distributions) which is related to the idea that the measurement of physical quantities by devices is based
on averaging,
• global symmetry and conservation laws,
• local symmetry and the basic ideas of modern gauge field theory, and
• the Planck quantum of action and the idea of quantizing classical field theories.

Gauge field theory is behind both
• the Standard Model in elementary particle physics and
• Einstein’s theory of gravitation (i.e., the theory of general relativity).

In quantum field theory, a crucial role is played by renormalization. In terms
of physics, this is based on the following two steps:
• the regularization of divergent integrals, and
• the computation of effective physical parameters measured in experiments (e.g.,
the effective mass and the effective electric charge of the electron).

Renormalization is a highly technical subject. For example, the full proof
on the renormalizability of the electroweak sector of the Standard Model in
particle physics needs 100 pages. This can be found in:
E. Kraus, Renormalization of the electroweak standard model to all orders,
Annals of Physics 262 (1998), 155–259.

Whoever wants to understand quantum field theory has to understand the
procedure of renormalization. Therefore, the different aspects of renormalization theory will be studied in all of the six volumes of this series of monographs. This ranges from
• resonance phenomena for the anharmonic oscillator (classical bifurcation theory),
• the Poincar´e–Lindstedt series (including small divisors) in celestial mechanics,



X

Preface

• and the Kolmogorov–Arnold–Moser (KAM) theory for perturbed quasi-periodic
oscillations (e.g., in celestial mechanics) based on sophisticated iterative techniques (the hard implicit function theorem)

to the following fairly advanced subjects:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•

the Feynman functional integral (the Faddeev–Popov approach),
the Wiener functional integral (the Glimm–Jaffe approach),
the theory of higher-dimensional Abelian integrals (algebraic Feynman integrals),
Hopf algebras and Rota–Baxter algebras in combinatorics (the modern variant of the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach due to
Kreimer),

the Riemann–Hilbert problem and the Birkhoff decomposition (the Connes–
Kreimer approach),
Hopf superalgebras (the Brouder–Fauser–Frabetti–Oeckl (BFFO) approach),
characterization of physical states by cohomology and algebraic renormalization
(the Becchi–Rouet–Stora–Tyutin (BRST) approach),
the Riesz–Gelfand theory of distribution-valued meromorphic functions (construction of the Green’s functions),
wave front sets and Hă
ormanders multiplication of distributions (the Stueckelberg
BogoliubovEpsteinGlaserScharf approach),
the Master Ward identity as a highly non-trivial renormalization condition and
the generalized DysonSchwinger equation (the Dă
utschFredenhagen approach),
q-deformed quantum eld theory (the Wess–Majid–Wachter–Schmidt approach
based on the q-deformed Poincar´e group, quantum groups, and the q-analysis on
specific classes of q-deformed quantum spaces),
deformation of bundles and quantization (the Weyl–Flato–Sternheimer–Fedosov–
Kontsevich approach),
microlocal analysis and renormalization on curved space-times (the Radzikowski
BrunettiFredenhagenKă
ohler approach),
renormalized operator products on curved space-times (the Wilson–Hollands–
Wald approach to quantum field theory),
natural transformations of functors in category theory and covariant quantum
field theory on curved space-time manifolds (the Brunetti–Fredenhagen–Verch
approach),

as well as
• one-parameter Lie groups and the renormalization group,
• attractors of dynamical systems in the space of physical theories (the Wilson–
Polchinski–Kopper–Rivasseau approach to renormalization based on the renormalization group),

• the Master Ward Identity and the Stueckelberg–Petermann renormalization
group (the Dă
utschFredenhagen approach),
ã motives in number theory and algebraic geometry, the Tannakian category, and
the cosmic Galois group as a universal (motivic) renormalization group (the
Connes–Marcolli approach),
• noncommutative geometry and renormalization (the Grosse–Wulkenhaar approach).

The recent work of Alain Connes, Dirk Kreimer, and Matilde Marcolli shows
convincingly that renormalization is rooted in highly nontrivial mathematical structures. We also want to emphasize that the theory of many-particle
systems (considered in statistical physics and quantum field theory) is deeply
rooted in the theory of operator algebras. This concerns


Preface
•
•
•
•

XI

von Neumann algebras (the von Neumann approach),
C ∗ -algebras (the Gelfand–Naimark–Segal approach),
local nets of operator algebras (the Haag–Kastler approach) and,
noncommutative geometry (the Connes approach).

As a warmup, we show in Chapter 2 that the regularization of divergent
expressions represents a main subject in the history of mathematics starting
with Euler in the eighteenth century. In this connection, we will consider

• the regularization of divergent series, and
• the regularization of divergent integrals.

In particular, in Sect. 2.1.3, we will discuss the classical Mittag–Leffler theorem on meromorphic functions f . If the function f has merely a finite number
of poles, then the method of partial fraction decomposition works well. However, as a rule, this method fails if the function f has an infinite number of
poles. In this case, Mittag–Leffler showed in the late nineteenth century that
one has to subtract special terms, which are called subtractions by physicists.
The subtractions force the convergence of the infinite series.
This is the prototype of the method of iteratively adding subtractions in the
Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach to renormalization theory. The corresponding iterative algorithm (called the Bogoliubov
R-operation) has to be constructed carefully (because of nasty overlapping
divergences). This was done by Nikolai Bogoliubov in the 1950s. An ingenious
explicit solution formula for this iterative method was found by Wolfhart Zimmermann in 1969. This is the famous Zimmermann forest formula. In the
late 1990s, it was discovered by Dirk Kreimer that the sophisticated combinatorics of the Zimmermann forest formula can be understood best in terms
of a Hopf algebra generated by Feynman diagrams. By this discovery, the
modern formulation of the BPHZ approach is based on both Hopf algebras
and Rota–Baxter algebras.
As a warmup, in Chapter 3, we give an introduction to the modern combinatorial theory, which was founded by Gian-Carlo Rota (MIT, Cambridge,
Massachusetts) in the 1960s. This includes both Hopf algebras and Rota–
Baxter algebras.
Surprisingly enough, it turns out that the Zimmermann forest formula is closely related to methods developed by Lagrange in the eighteenth century when studying the solution of the Kepler equation for
the motion of planets in celestial mechanics.
In modern terminology, the Lagrange inversion formula for power series expansions is based on the so-called Fa`a di Bruno Hopf algebra.1 This will be
studied in Sect. 3.4.3.
1

The Italian priest and mathematician Francesco Fa`
a di Bruno (1825–1888) was
beatified in 1988.



XII

Preface

In physics, symmetries are basic. For describing symmetries in terms of
mathematics, there are two approaches based on
• groups and
• Hopf algebras.

In 1941, Heinz Hopf wanted to compute the cohomology of topological groups.
Hopf discovered that the cohomology ring of topological groups is equipped
with an additional structure which is called a Hopf algebra structure today.
This additional algebraic structure is based on the notion of the coproduct.
Roughly speaking, the concept of Hopf algebra is dual to the concept of
group. Hopf algebras are intimately related to quantum groups. We will show
in Chapter 3 that:
The product and the coproduct of a Hopf algebra model the fusion
and the splitting of objects (e.g., elementary particles), respectively.
In terms of analysis, the algebra of linear differential operators with constant
coefficients can be equipped with the structure of a Hopf algebra. Here,
• the coproduct is related to the Leibniz product rule of differentiation, and
• the coinverse (also called the antipode) is related to the integration-by-parts
formula (see Sect. 3.3.1).

The integration-by-parts formula is a special case of the general Stokes integral theorem, which lies at the heart of the duality between homology and
cohomology in topology. This duality plays a key role for the mathematical
description of processes in nature. In particular, cohomology is deeply rooted
in Maxwell’s theory of electrodynamics (see Sect. 4.4.7).
Incredible cancellations. When doing computations in renormalization

theory, as a big surprise, physicists and mathematicians encounter incredible
cancellations of a large amount of terms. This dramatically simplifies the
final result. In terms of mathematics, a sophisticated combinatorics is behind
these cancellations. The prototype for this is given by the Fa`
a di Bruno Hopf
algebra mentioned above.
The language of modern mathematics. We do not assume that the
reader of this series of monographs is familiar with the language used in modern mathematics. In this connection, we want to help the reader. For example,
many notions in advanced mathematics and hence in modern mathematical
physics are based on mathematical operations applied to equivalence classes.
For example, this concerns
• the construction of quantum states as equivalence classes of elements of Hilbert
spaces (and the relation to projective geometry),
• the Gelfand–Naimark–Segal (GNS) construction for representing the elements of
an abstract C ∗ -algebra as observables on a Hilbert space (the algebraic approach
to quantum theory),
• the Wightman reconstruction theorem for axiomatically defined quantum fields
(via the GNS-construction),


Preface

XIII

• moduli spaces of Riemann surfaces (modulo conformal equivalence) and physical
states in string theory.

This leads to quotient spaces in algebra, analysis, geometry, and topology,
which will be encountered again and again in this series of monographs (e.g.,
homology groups, cohomology groups, homotopy groups, and K-theory in

topology). Chapter 4 serves as an introduction to quotient structures in mathematics and physics. The idea of the quotient ring (modulo a fixed integer)
can be traced back at least to the Disquisitiones arithmeticae written by the
young Gauss (1777–1855) in 1801.2 In order to give the reader a feel for the
usefulness of working with equivalence classes, we will consider the following
examples:
• Gaussian quotient rings (modulo a prime number) and coding theory (as warmup
for quantum information),
• quotient fields and Heaviside’s symbolic method in electrical engineering (the
Mikusi´
nski operational calculus),
• physical fields, observers, bundles, and cocycles,
• deformation, mapping classes, and topological charges,
• loop spaces and higher homotopy groups,
• the projective and the injective limit of mathematical structures (e.g., topological
spaces), and
• the rigorous approach to Leibniz’s infinitesimals via ultrafilters (nonstandard
analysis).

For the foundation of nonstandard analysis, one needs the construction of
ultrafilers via Zorn’s lemma based on the axiom of choice in set theory.
The language of theoretical physics. Chapters 5 through 9 are devoted
to the basic ideas of
• classical geometric optics,
• classical mechanics, and
• quantum mechanics.
Here, we want to help mathematicians who are not familiar with theoretical
physics. In Chapter 5, we study Carath´eodory’s royal road to geometrical
optics based on the fundamental duality between
• light rays and
• wave fronts


which can be traced back to the work of Huygens in the seventeenth century. In string theory, Kă
ahler manifolds play a crucial role. In Chapter 5, we
will show how Poincar´e’s non-Euclidean geometry on the upper half-plane is
related to both geometrical optics and Kă
ahler geometry.
2

The enormous inuence of Gauss’ first masterpiece on the development of mathematics is described in the monograph by C. Goldstein, N. Schappacher, and J.
Schwermer: The Shaping of Arithmetic after Gauss’ Disquisitiones Arithmeticae,
Springer, Berlin 2007.


XIV

Preface

Since all the models of quantum fields are based on the study of an infinite
number of (slightly perturbed) harmonic oscillators in the setting of perturbation theory, we use the harmonic oscillator as a paradigm for both classical
mechanics and quantum mechanics. In Chapter 6 on classical mechanics, we
will study the following topics:
• Newtonian mechanics,
• Lagrangian mechanics (the Euler–Lagrange equation, the Jacobi accessory eigenvalue problem and Morse theory),
• Hamiltonian mechanics (the canonical dynamical system and the Hamilton–
Jacobi partial differential equation), and
• Poissonian mechanics.

In particular, this concerns
• the Legendre transformation and contact geometry,
• the Hamiltonian flow and symplectic geometry,

• the tangent bundle of the position space (the position-velocity space also called
the state space),
• the cotangent bundle of the position space (the position-momentum space also
called the phase space),
• the Legendre transformation as a transformation from the tangent bundle to the
cotangent bundle; the latter is equipped with a natural symplectic structure.

In terms of mathematics, the fundamental relation between symmetry and
conservation laws in physics is related to
• the Noether theorem, and
• Poisson brackets and Lie’s momentum map.

Quantum mechanics. The comprehensive Chapter 7 lies at the heart of
this series of monographs. This chapter should help the reader to understand
the different aspects of the passage from classical physics to quantum physics,
by using the different procedures of quantization. We will use the paradigm
of the harmonic oscillator in order to explain the basic ideas of the following
approaches:
• Heisenberg’s quantum mechanic (via creation and annihilation operators),
ã Schră
odingers quantum mechanics (via the Schră
odinger partial dierential equation),
ã Feynmans quantum mechanics (via the path integral),
• von Neumann’s functional-analytic approach (via the spectral theory for selfadjoint operators in Hilbert spaces),
• von Neumann’s density operator in statistical physics (via trace class operators),
• Weyl’s symbolic calculus for pseudo-differential operators (deformation quantization),
• the Poincar´e–Wirtinger calculus and Bargmann’s holomorphic quantization,
• the Stone-von Neumann uniqueness theorem (for the fundamental commutation
relations in quantum mechanics) and the Weyl functor3 based on symplectic
geometry,

• supersymmetric quantization.
3

At this place, the general theory of mathematical structures (also called category
theory) enters the theory of quantization (also called quantum mathematics).


Preface

XV

Concerning the Feynman path integral as a fundamental tool in quantum
physics, we will study the following:
• Brownian motion and the infinite-dimensional rigorous Wiener integral based on
measure theory,
• the rigorous Feynman–Kac formula for diffusion processes,
• rigorous finite-dimensional Gaussian integrals, the computation of correlations
and moments, the Wick theorem, and Feynman diagrams,
• rigorous definition of infinite-dimensional Gaussian integrals via zeta function
regularization,
• the Wentzel–Kramers–Brioullin (WKB) method of stationary phase for the computation of Gaussian integrals, and the approximate computation of Feynman
path integrals.

The Feynman path integral can be obtained from the Wiener integral by
using formal analytic continuation from real time to imaginary time. This
corresponds to the fact that the Schră
odinger equation describes diusion
processes in imaginary time. Furthermore, in Chapter 7, we discuss the basic
ideas of the algebraic approach to quantum mechanics by using C ∗ -algebras
and von Neumann algebras. In this connection, we consider:

• applications to statistical mechanics (Boltzmann statistics, Bose–Einstein statistics, and Fermi–Dirac statistics),
• thermodynamic equilibrium states (Kubo–Martin–Schwinger (KMS) states) and
the Tomita–Takesaki theory for von Neumann algebras,
• the Murray–von Neumann classification of factors in the theory of von Neumann
algebras,
• projection operators and the main theorem of quantum logic (Gleason’s extension
theorem for C ∗ -algebras).

The modern theory of operator algebras culminates in Alain Connes’s noncommutative geometry, which represents the appropriate mathematical structure for a deeper understanding of the Standard Model in elementary particle
physics. This will be investigated in Volume IV on quantum mathematics.
For the interested reader, we refer to the following fundamental monograph:
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields,
and Motives, American Mathematical Society 2008.
Internet: marcolli/bookjune4.pdf

Chapters 8 and 9 serve as an introduction to scattering theory, which plays
a crucial role in elementary particle physics. As a paradigm for general scattering theory, we consider the scattering of a quantum particle on the real
line. We consider
• the energy levels of bound states,
• the energy levels of scattering states, and distributions as generalized eigenfunctions of the Schră
odinger equation,
ã the transition matrix,
ã the unitary S-matrix and transition probabilities for scattering processes,
• the relation between the singularities of the S-matrix in the complex energetic
plane and the energy levels of stable bound states,


XVI

Preface


• unstable particles (resonances) and the second sheet of the energetic Riemann
surface (the Breit–Wigner formula for the energy levels and the mean lifetime of
resonances),
• stationary scattering theory, the Green’s function of the Helmholtz equation, and
the Lippmann–Schwinger integral equation,
• instationary scattering theory, wave operators, the absolutely continuous spectrum of the Hamiltonian, and the S-matrix in functional analysis.

Here, we do not assume that the reader is familiar with
• von Neumann’s functional-analytic spectral theory for self-adjoint operators in
Hilbert spaces,
• the Gelfand–Kostyuchenko theory of generalized eigenfunctions for self-adjoint
operators,
• the Møller–Kato theory of wave operators in scattering theory, and
• the Weyl–Kodaira theory for singular differential operators.

For the convenience of the reader, the necessary material will be summarized
at the proper place when it is needed in Volumes II and III.
Quantum electrodynamics. In the present volume, it is our main goal to
illustrate the beauty of quantum electrodynamics by proceeding pragmatically.
We do not start with an abstract approach, but with the computation of important physical effects which are observed in experiments,
including radiative corrections in lowest order of renormalization theory.
This should help the reader in getting a feel for the essential questions. More
sophisticated approaches are postponed to later volumes of this series of
monographs. In the introductory Chapter 10, we study creation and annihilation operators for electrons, positrons, and photons. In Chapter 11, we
formulate the classical field equations of quantum electrodynamics on the interaction between electrons and photons, by coupling the Maxwell equations
of the electromagnetic field to the Dirac equation of the electron wave function. This equation depends on the gauge fixing of the four-potential for the
electromagnetic field. However, it turns out that physical effects measured in
experiments are independent of the choice of the gauge fixing. The point is
that:

The classical field equations of quantum electrodynamics have to be
quantized.
In this connection, we have to distinguish between
• the single free quantum fields for electrons, positrons, and photons, and
• the total interacting quantum field for electrons, positrons, and photons.

In Chapter 12, we construct free quantum fields by using the method of
Fourier quantization based on creation and annihilation operators.
In Chapter 13, we study the interacting quantum field of electrons,
positrons, and photons by using


Preface

XVII

• the magic Dyson formula for the S-matrix (scattering matrix), and
• the main Wick theorem for the S-matrix, which implies the Feynman diagrams.

This is Dyson’s classical approach to understanding the Feynman diagrams.4
Originally, Feynman invented his exciting diagram technique on the basis of
ingenious physical intuition. In Dyson’s mathematical setting, the Feynman
diagrams are nothing other than graphical representations of well-defined analytic expressions, which are effectively produced by the main Wick theorem.
Feynman’s use of propagators and Dyson’s magic formula for the S-matrix
are closely related to Lagrange’s variation-of-parameter formula in celestial
mechanics. Many mathematicians complain about the following situation:
In the physics textbooks, one reads the Feynman rules for Feynman
diagrams, but it is not clear where the Feynman rules come from.
In the present textbook, we will thoroughly study the mathematical and
physical origin of both the Feynman diagrams and the Feynman rules. We

will also consider applications to interesting physical effects.
In Chapter 15, we investigate the following physical effects in lowest order
of perturbation theory:
• the cross section for Compton scattering between photons and electrons (improvement of the Thomson formula in classical electrodynamics),
• the cross section for the scattering of electrons in an external electromagnetic
field,
• the intensity of spectral lines for bound states in an external electromagnetic
field, and
• the Cherenkov radiation.

For the computation of terms corresponding to higher order of perturbation theory, renormalization theory is needed. In Chapter 17, we discuss the
physics behind the following radiative corrections in lowest possible order of
renormalization theory:
• the screening of the Coulomb potential by vacuum polarization (the Uehling
potential),
• the anomalous magnetic moment of the electron (the Schwinger formula),
• the anomalous magnetic moment of the muon, and
• the Lamb shift in the spectrum of the hydrogen atom.

Unfortunately, the explicit computations (in the framework of dimensional
regularization in renormalization theory) are lengthy. We will postpone these
detailed computations to Volume III.
In Chapter 18, we discuss the main result telling us that quantum electrodynamics can be renormalized to all orders of perturbation theory. The
final result consists of getting finite expressions in each order of perturbation
theory (e.g., cross sections for scattering processes), which depend on the two
fundamental free parameters
4

Dyson’s discovery of this approach is described by himself in his book, F. Dyson,
Disturbing the Universe, Harper & Row, New York, 1979 (see page 27 of Volume

I for this fascinating story).


XVIII

Preface

• meff (effective mass of the electron) and
• −eeff (effective electric charge of the electron).

Observe the crucial fact that:
The free parameters meff and eeff cannot be determined theoretically
by quantum electrodynamics.
They have to be determined by physical experiments. In the SI system, one
obtains the following values:
meff = 0.511 MeV/c2 ,

eeff = 1.602 · 10−19 As.

A reader who wants to become familiar with quantum electrodynamics as
quickly as possible should start reading with Chapter 10.
The incredible effectiveness of perturbation theory in physics. Surprisingly enough, low-order radiative corrections are sufficient for getting a
fantastic coincidence between theory and experimental data. For example,
the anomalous magnetic moment of the electron measured in experiments is
predicted very precisely by fourth-order radiative corrections (up to 9 digits).
However, the necessary amount of computations is enormous. One has to evaluate high-dimensional integrals which correspond to 891 Feynman diagrams;
this needs years of supercomputer time.
A warning to the reader. In summer 1976, Arthur Wightman (Princeton
University) organized a famous conference in Erice (Sicily/Italy) on renormalization theory. In the introduction to the Proceedings of this conference,
he writes:5

Renormalization theory is a notoriously complicated and technical subject. . . I want to tell stories with a moral for the earnest student: Renormalization theory has a history of egregious errors by distinguished savants
(see page 967). It has a justified reputation for perversity; a method that
works up to 13th order in the perturbation theory fails in the 14th order.
Arguments that sound plausible often dissolve into mush when examined
closely. The worst that can happen often happens. The prudent student
would do well to distinguish sharply between what has been proved and
what has been plausible, and in general he should watch out!

In 1999 Gerardus ’t Hooft and Martinus Veltman were awarded the Nobel
prize in physics for their contributions to the renormalization of the theory
of electroweak interaction and for the computation of radiative corrections in
the Standard Model of particle physics.
Perspectives. More advanced approaches to renormalization theory will be
systematically studied in the following volumes of this series of monographs.
In particular, this concerns the new approaches to perturbative quantum field
theory due to Connes and Kreimer (Hopf algebras), and Brunetti, Dă
utsch,
5

A. Wightman, Orientation. In: Renormalization Theory, pp. 1–20. Edited by G.
Velo and A. Wightman, Reidel, Dordrecht, 1976 (reprinted with permission).


Preface

XIX

and Fredenhagen (microlocal analysis and the Master Ward Identity). In
order to give the reader an overview on the large variety of different approaches to renormalization theory, we summarize important references in
Section 19.11, and in Chapter 19 we sketch some basic ideas.

The propagation of light in the universe, namely,
• the deflection of light at the sun, and
• the red shift of spectral lines as a consequence of the expansion of the universe
(the Hubble effect)

will be investigated in Volume III in terms of Einstein’s theory of general
relativity.
The basic idea of our approach to quantum electrodynamics. As a
rule, mathematicians have trouble with reading some textbooks on quantum
field theory written by physicists. The point is that:
In mathematics, one never does computations with quantities which
do not exist.6
In order to respect this basic principle in mathematics, we will use the lattice
approach. That is, roughly speaking, we will proceed in the following two
steps.
Step 1: The discretized physical system. We put the physical system in a cubic
box of finite side length L and volume V = L3 . The boundary conditions are
given by periodicity.
• We observe the physical system in a finite time interval [− T2 , T2 ].
• We choose a maximal energy Emax .
• In the 3-dimensional momentum space, we introduce a finite lattice of spacing
Δp and maximal momentum Pmax . In this setting, the Fourier integral transform
is replaced by a discrete Fourier transform via finite Fourier series expansions.
• We define Dyson’s S-matrix for this situation.
• The main Wick theorem allows us to compute the S-matrix elements (i.e., the
transition amplitudes) in an elegant manner, by eliminating the creation and
annihilation operators, and replacing them by propagators (i.e., correlation functions for free fields).
• The point is that the propagators are discrete algebraic Feynman integrals, which
are indeed well-defined finite sums.
• The transition amplitudes can be graphically represented by Feynman diagrams.

• The Feynman rules allow us to translate the Feynman diagrams into well-defined
finite sums.
• From the transition amplitudes, we obtain the transition probabilities which
yield the cross sections for scattering processes. Note that cross sections can be
measured in particle accelerators.

Step 2: The delicate continuum limit. Explicitly, we have to study the following limits:
• L → ∞ (the volume L3 of the cubic box becomes infinite),
6

For example, this concerns infinite renormalization constants or ill-defined
infinite-dimensional functional/path integrals.


XX

Preface

• T → ∞ (the time interval becomes infinite),
• Pmax → ∞ (i.e., Emax → ∞) (high-energy limit),
• Δp → 0 (low-energy limit).
In order to force the convergence of the discrete algebraic Feynman integrals
to well-defined expressions, we modify the classical Lagrangian density by
setting
eeff := e + δe.
meff := me + δm,
That is, we replace the so-called bare electron mass me and the so-called bare
electron charge −e in the Lagrangian density by
me = meff − δm,


−e = −eeff + δe,

respectively. This way, the classical Lagrangian density
L(ψ, ∂ψ, A, ∂A; me , e)
is modified by the function
Lmodified (ψ, ∂ψ, A, ∂A; meff , eeff ; δm, δe).
The terms multiplied by δm, δe are called counterterms of the classical Lagrangian density L. Note that in this lattice approach, δe and δm are real
parameters which depend on the shape of the lattice, that is, they depend on
the maximal energy Emax . Now consider the high-energy limit
Emax → +∞.
Roughly speaking, we have to show that δm(Emax ) and δe(Emax ) can be
chosen in such a way that the finite continuum limit exists for the S-matrix
elements (i.e., the transition elements). This is the procedure of renormalization.
Observe the following peculiarity. By the Stone–von Neumann uniqueness
theorem, a finite number of creation and annihilation operators is uniquely
determined by the commutation relations (up to unitary equivalence). This is
not true anymore for an infinite number of creation and annihilation operators, as was shown by Lars G˚
arding and Arthur Wightman in 1954. However,
our approach avoids the latter ambiguity, since we only work with a finite
number of creation and annihilation operators before passing to the continuum limit (of the vacuum expectation values). We also would like to emphasize that our approach differs only slightly from the usual approach used by
physicists. In particular, we use a notation for discrete Fourier integrals such
that the formal passage to the language used by physicists is possible at each
stage of our procedure.
For the physical quantities which can be measured in experiments,
our final formulas coincide with the formulas used by physicists.


Preface

XXI


Moreover, in each step of our procedure it is easy to pass formally to the
expressions used by physicists, since the Feynman diagrams are the same.
This way, we hope to help mathematicians in getting a better understanding for the ingenious and beautiful approach invented by physicists.
From the physical point of view, the modification of the classical Lagrangian
density reflects the fact that:
Quantum effects have to be added to the classical theory.
Intuitively, this means that the quantum fluctuations of the ground state
of the quantum field of electrons, positrons, and photons influence physical
effects observed in experiments. For example, this concerns the anomalous
magnetic moment of the electron and the spectrum of the hydrogen atom
(Lamb shift).
Convention. If we do not expressively state the opposite, we will use the SI
system of physical units (international system of units) which can be found
in the Appendix to Volume I. In particular, note that in Chapters 10–19
on quantum electrodynamics, we will use the energetic system with c = 1
(velocity of light in a vacuum), = h/2π = 1 (Planck’s quantum of action),
k = 1 (Boltzmann constant), ε0 = μ0 = 1 (see page 790).
The Poincar´
e Seminar. The best way of getting information about recent developments in modern physics is to look at the books which report
the lectures given at the Poincar´e Seminar in Paris. Starting in 2002, this
seminar has been organized by l’Institut Henri Poincar´e in Paris (see page
1050). Bertrand Duplantier and Vincent Rivasseau write in the Foreword to
Quantum Spaces, Birkhă
auser, Basel, 2007:
This book is the seventh in a series of lectures of the S´eminaire Poincar´e,
which is directed towards a large audience of physicists and mathematicians.
The goal of this seminar is to provide up-to-date information about general topics of great interest in physics. Both the theoretical and experimental aspects are covered, with some historical background. Inspired by the
S´eminaire Bourbaki in mathematics in its organization, hence nicknamed
“S´eminaire Bourbaphy,” the Poincar´e Seminar is held twice a year at the

Institut Henri Poincar´e in Paris, with contributions prepared in advance.
Particular care is devoted to the pedagogical nature of the presentation so
as to fulfill the goal of being readable by a large audience of scientists.

Two recent survey volumes. The following two volumes try to reflect
the state of the art by summarizing the most important approaches used in
modern quantum field theory:
• B. Fauser, J. Tolksdorf, and E. Zeidler (Eds.), Quantum Gravity: Mathematical
Models and Experimental Bounds, Birkhă
auser, Basel, 2006.
• B. Fauser, J. Tolksdorf, and E. Zeidler (Eds.), Quantum Field Theory Competitive Methods, Birkhă
auser, Basel, 2008.


XXII

Preface

Acknowledgements. I am very grateful to my collaborators Bertfried
Fauser and Jă
urgen Tolksdorf for organizing two workshops in 2005 and 2007
on recent developments in quantum field theory, and for both doing an excellent editorial job and telling me new research strategies and new results
in the algebraic and geometric approach to quantum field theory. In particular, I would like to thank Bertfried Fauser for drawing my attention to the
relation between quantum field theory and the following topics: Gian-Carlo
Rota’s work on combinatorics, the ubiquitous Hopf algebras, Rota–Baxter
algebras, quantum Clifford algebras, the Rota–Stein cliffordization process,
Littlewood’s work on the representation theory of the symmetric group, motivic number theory, topos theory, and category theory in algebraic geometry.
For illuminating discussions about promising new approaches to (perturbative) quantum field theory in both flat and curved space-time, I would like
to thank Klaus Fredenhagen and the late Julius Wess, as well as Detlev
Buchholz, Christian Brouder, Romeo Brunetti, Michael Dă

utsch, Kurusch
Ebrahimi-Fard, Joel Feldman, Felix Finster, Christian Fleischhack, Alessandra Frabetti, Stefan Hollands, Harald Grosse, Jerzy Kijowski, Dirk Kreimer,
Elisabeth Kraus, Alexander Lange, Matilde Marcolli, Mario Paschke, Klaus
Rehren, Gerd Rudolph, Manfred Salmhofer, Alexander Schmidt, Klaus Sibold, Rainer Verch, Hartmut Wachter, and Raimar Wulkenhaar.
I would like to thank Thomas Hahn from the Max Planck Institute for
Physics, Werner Heisenberg, in Munich for informing me on the state of the
art in automated multi-loop computations in perturbation theory. Such sophisticated computer programs are used for preparing the experiments at the
LHC (Large Hadron Collider) of CERN (European Organization for Nuclear
Research at Geneva, Switzerland) (see Sect. 18.4).
On the occasion of Professor Friedrich Hirzebruch’s 80th birthday on October 17 in 2007, I would like to dedicate this volume to him in gratitude. His
scientific work deeply influenced the development of mathematics in the second half of the twentieth century. Nowadays physicists frequently use Hirzebruch’s results in order to study the topological structure of physical fields.
In 1982, Friedrich Hirzebruch founded the Max Planck Institute for Mathematics in Bonn (Germany). Mathematicians from all over the world enjoy
doing research in the relaxed and highly stimulating atmosphere of this institute. In 1996, the Max Planck Institute for Mathematics in the Sciences was
founded in Leipzig. Friedrich Hirzebruch was the chairman of the Founders’
Committee. The staff and the visitors of our institute are very grateful to
Professor Hirzebruch for his efforts made as chairman.
For helping me to save a lot of time, I am very grateful to my secretary
Regine Lă
ubke (invaluable support), Katarzyna Baier (answering patiently
almost innitely many bibliographical questions), the library team (steadily
support), Kerstin Fă
olting (graphics and tables), Micaela Krieger-Hauwede
(answering patiently my LATEX questions), Katrin Scholz (internet searching),
and Thomas Heid (computer expert). I also would like to thank the staff of


Preface

XXIII


the Springer publishing house in Heidelberg, Ruth Allewelt, Joachim Heinze,
and Martin Peters, for the harmonious collaboration. Many years ago, my
Czech colleague from Prague, the late Svatopluk Fuˇcik, wrote the following
sentence in the preface to his book on nonlinear partial differential equations:
“Finally, I would like to thank myself for typing the manuscript.” I resist the
temptation of repeating this sentence with respect to my LATEX file. But I
would like to thank Donald Knuth for enjoying the mathematical community
with the beautiful gift of the TEX tool.
In the Proverbs 31, 10 of the Bible, one reads:
Who can find a virtuous woman? For her price is far above rubies. The
heart of her husband does safely trust in her, so that he shall have no need
of spoil. She will do him good and not evil all the days of her life.

I am very grateful to my beloved wife, Christine, who has been taking care
of me for 40 years.
Leipzig, Summer 2008

Eberhard Zeidler


Contents

Part I. Introduction
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.

2.

Mathematical Principles of Modern Natural Philosophy .
1.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 The Infinitesimal Strategy and Differential Equations . . . . . .
1.3 The Optimality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 The Basic Notion of Action in Physics and the Idea of
Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 The Method of the Green’s Function . . . . . . . . . . . . . . . . . . . .
1.6 Harmonic Analysis and the Fourier Method . . . . . . . . . . . . . .
1.7 The Method of Averaging and the Theory of Distributions .
1.8 The Symbolic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Gauge Theory – Local Symmetry and the Description of
Interactions by Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 The Challenge of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . .
The Basic Strategy of Extracting Finite Information from
Infinities – Ariadne’s Thread in Renormalization Theory
2.1 Renormalization Theory in a Nutshell . . . . . . . . . . . . . . . . . . .
2.1.1 Effective Frequency and Running Coupling Constant
of an Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . .
2.1.2 The Zeta Function and Riemann’s Idea of Analytic
Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Meromorphic Functions and Mittag-Leffler’s Idea
of Subtractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 The Square of the Dirac Delta Function . . . . . . . . . . .
2.2 Regularization of Divergent Integrals in Baby
Renormalization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Momentum Cut-off and the Method of
Power-Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The Choice of the Normalization Momentum . . . . . .
2.2.3 The Method of Differentiating Parameter Integrals .
2.2.4 The Method of Taylor Subtraction . . . . . . . . . . . . . . .

1

11
12
14
14
15
17
21
26
28
34
46

47
48
48
54
56
58
60
60
63
63
64


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Contents

2.2.5

2.2.6
2.2.7
2.2.8
2.2.9

2.3

2.4

2.5
3.

The
3.1
3.2
3.3

3.4

Overlapping Divergences . . . . . . . . . . . . . . . . . . . . . . . .
The Role of Counterterms . . . . . . . . . . . . . . . . . . . . . . .
Euler’s Gamma Function . . . . . . . . . . . . . . . . . . . . . . . .
Integration Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dimensional Regularization via Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.10 Pauli–Villars Regularization . . . . . . . . . . . . . . . . . . . . .
2.2.11 Analytic Regularization . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.12 Application to Algebraic Feynman Integrals in
Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.13 Distribution-Valued Meromorphic Functions . . . . . . .
2.2.14 Application to Newton’s Equation of Motion . . . . . .

2.2.15 Hints for Further Reading. . . . . . . . . . . . . . . . . . . . . . .
Further Regularization Methods in Mathematics . . . . . . . . . .
2.3.1 Euler’s Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Adiabatic Regularization of Divergent Series . . . . . . .
2.3.3 Adiabatic Regularization of Oscillating Integrals . . .
2.3.4 Regularization by Averaging . . . . . . . . . . . . . . . . . . . . .
2.3.5 Borel Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.6 Hadamard’s Finite Part of Divergent Integrals . . . . .
2.3.7 Infinite-Dimensional Gaussian Integrals and the Zeta
Function Regularization . . . . . . . . . . . . . . . . . . . . . . . . .
Trouble in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Interchanging Limits . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 The Ambiguity of Regularization Methods . . . . . . . .
2.4.3 Pseudo-Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Ill-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathemagics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power of Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Algebra of Multilinear Functionals . . . . . . . . . . . . . . . . . .
Fusion, Splitting, and Hopf Algebras . . . . . . . . . . . . . . . . . . . .
3.3.1 The Bialgebra of Linear Differential Operators . . . . .
3.3.2 The Definition of Hopf Algebras . . . . . . . . . . . . . . . . .
Power Series Expansion and Hopf Algebras . . . . . . . . . . . . . . .
3.4.1 The Importance of Cancellations . . . . . . . . . . . . . . . . .
3.4.2 The Kepler Equation and the Lagrange
Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 The Composition Formula for Power Series . . . . . . . .
3.4.4 The Fa`
a di Bruno Hopf Algebra for the Formal
Diffeomorphism Group of the Complex Plane . . . . . .

3.4.5 The Generalized Zimmermann Forest Formula . . . . .
3.4.6 The Logarithmic Function and Schur Polynomials . .
3.4.7 Correlation Functions in Quantum Field Theory . . .

65
67
67
69
73
76
77
80
81
87
92
93
93
94
95
96
98
100
101
102
102
104
104
105
109
115

115
117
122
123
128
131
131
132
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141


Contents XXVII

3.4.8 Random Variables, Moments, and Cumulants . . . . . .
Symmetry and Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 The Strategy of Coordinatization in Mathematics
and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 The Coordinate Hopf Algebra of a Finite Group . . .
3.5.3 The Coordinate Hopf Algebra of an Operator Group
3.5.4 The Tannaka–Krein Duality for Compact Lie Groups
Regularization and Rota–Baxter Algebras . . . . . . . . . . . . . . . .
3.6.1 Regularization of the Laurent Series . . . . . . . . . . . . . .
3.6.2 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 The q-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.4 The Volterra–Spitzer Exponential Formula . . . . . . . .
3.6.5 The Importance of the Exponential Function in

Mathematics and Physics . . . . . . . . . . . . . . . . . . . . . . .
Partially Ordered Sets and Combinatorics . . . . . . . . . . . . . . . .
3.7.1 Incidence Algebras and the Zeta Function . . . . . . . . .
3.7.2 The Mă
obius Function as an Inverse Function . . . . . .
3.7.3 The Inclusion–Exclusion Principle in Combinatorics
3.7.4 Applications to Number Theory . . . . . . . . . . . . . . . . . .
Hints for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143
146

The Strategy of Equivalence Classes in Mathematics . . . . .
4.1 Equivalence Classes in Algebra . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Gaussian Quotient Ring and the Quadratic
Reciprocity Law in Number Theory . . . . . . . . . . . . . .
4.1.2 Application of the Fermat–Euler Theorem in Coding
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Quotient Rings, Quotient Groups, and Quotient Fields
4.1.4 Linear Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.5 Ideals and Quotient Algebras . . . . . . . . . . . . . . . . . . . .
4.2 Superfunctions and the Heaviside Calculus in Electrical
Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Equivalence Classes in Geometry . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The Basic Idea of Geometry Epitomized by Klein’s
Erlangen Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Symmetry Spaces, Orbit Spaces, and Homogeneous
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 The Space of Quantum States . . . . . . . . . . . . . . . . . . .
4.3.4 Real Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.5 Complex Projective Spaces . . . . . . . . . . . . . . . . . . . . . .
4.3.6 The Shape of the Universe . . . . . . . . . . . . . . . . . . . . . .
4.4 Equivalence Classes in Topology . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Topological Quotient Spaces . . . . . . . . . . . . . . . . . . . . .
4.4.2 Physical Fields, Observers, Bundles, and Cocycles . .
4.4.3 Generalized Physical Fields and Sheaves . . . . . . . . . .

175
178

3.5

3.6

3.7

3.8
4.

146
148
150
152
154
157
158
158
160
161
162

162
163
164
166
167

178
182
184
188
190
191
194
194
194
199
200
203
204
205
205
208
216